This preprint is in final form. It appeared in:
J. Speidel (ed.), Neue Kommunikationsanwendungen in modernen Netzen
ITG-Fachbericht, Vol. 171, VDE Verlag, Berlin 2002, pp. 109–115
ISI/ICI COMPARISON OF DMT AND WAVELET BASED MCM SCHEMES FOR
TIME–INVARIANT CHANNELS
Maria Charina, Kurt Jetter, Achim Kehrein, Götz E. Pfander, and Georg Zimmermann
Institut für Angewandte Mathematik und Statistik, Universität Hohenheim, D–70593 Stuttgart, Germany
Werner Kozek
Siemens AG Österreich, Programm- und Systementwicklung, A–1030 Vienna, Austria
ABSTRACT
We present analytical results, numerical estimates, and
numerical simulations showing that wavelet based affine
MCM schemes are unfit for communication through
dispersive time invariant channels as given in DSL and
some wireless communication environments. Currently
used FFT based MCM schemes (DMT) outperform those
based on wavelets regardless of which wavelet is chosen.
1. INTRODUCTION
Modern digital communication systems show a general trend towards linear modulation schemes [1]. Important examples of linear modulation among others
are CDMA and OFDM (orthogonal frequency division
multiplex). The common principle of these schemes is
that the transmission signal can be written as a series
expansion based on a prescribed set of “pulses” which
are trigonometric polynomials in case of OFDM and
binary sequences in case of CDMA. Other variants are
the classical PAM or QAM modulation.
Any of these linear modulation schemes are part of
standardized digital communication systems, either for
wireless (e.g., HIPERLAN 2 in case of OFDM) or wired
(e.g. ADSL, where the baseband variant DMT of OFDM
is standardized for the asymmetrical transmission over
digital subscriber line) systems. We shall address one
of the key questions in the design of future communication systems concerning the design of the transmission
pulses in such a linear modulation scheme.
In principle any essentially time-limited and essentially
band-limited function can be taken as starting point for
the design of transmission signal sets. However, many
practical side constraints have to be fulfilled in order to
obtain realistic implementations and to enable the receiver to cope with the detrimental effects of the channel:
(i) The function system should be orthogonal, or at
least linearly independent, in order to have unique
demodulation.
(ii) The whole set of transmission pulses must have
a structure which admits fast DSP algorithms for
the implementation.
(iii) The function system must be robust with respect
to the distortions caused by the channel.
The introduction of wavelets almost 20 years ago had a
lasting effect on many fields in mathematics and engineering. For a detailed historical account see [5, 15]. In
information and communication theory, wavelets have
been successfully applied to source coding, e.g., JPEG
2000. Various authors have proposed the application of
wavelet systems for modulation/signal synthesis also,
e.g., see [4, 9, 16]. Even though wavelet families do
fulfill the first two constraints for multicarrier modulation (MCM) mentioned above, they fail the third objective: perturbation stability when faced with the dispersive effect caused by a linear translation invariant
operator. This failure leads to high ISI and ICI.
In this paper we shall present the problem of MCM
from a mathematical point of view and outline our mathematical/technical results concerning the “perturbation
stability of coherent Riesz-systems” [8] in Section 2
and Section 3. The emphasis of our proposed contribution is the comparison of shift invariant MCM systems
based on numerical simulations. We shall present exemplary results of those simulations in Section 5.
2. SHIFT INVARIANT MULTICARRIER
MODULATION
In linear multicarrier modulation systems the input signal is synthesized as a linear combination of basis functions {gi (t)}i∈I (I denotes a possibly infinite index
set), where the complex or real coefficients sustain the
information aimed at transmission. In the case of shift
invariant multicarrier modulation (Figure 1) the basis
functions are translates of a finite family of functions
{gl (t)}l=0,...,N −1 , i.e., for some T > 0 and
I = {0, . . . , N − 1} × Z we have
gl,k (t) = gl (t − kT ),
for l = 0, . . . , N − 1, k ∈ Z.
Hence, the resulting information carrying input signal
is given by
x(t) =
∞
X
k=−∞
xk (t) =
−1
∞ N
X
X
k=−∞ l=0
cl,k gl (t − kT ).
We assume throughout this work that the channel distortion due to transmission over a physical communication channel corresponds to a translation invariant system, i.e.,
Z
h(t−t′ ) x(t′ ) dt′ .
(Kh x)(t) = (h∗x)(t) =
R
c0,k -
c1,k
-
-
g0 (t − kT )
n(t)
x (t)
- Σ k - h(t)
-
g1 (t − kT )
r
r
r
cN −1,k gN −1 (t − ak)
?
- +
γ0 (t − kT )
c̃0,k
γ1 (t − kT )
c̃1,k
r
r
r
- γN −1 (t − kT )
c̃N −1,k
Fig. 1. Shift invariant multicarrier modulation
The transmitter strategy corresponds to a signal synthesis, accordingly the receiver performs a signal analysis,
in the sense of matched filtering by a family of functions with identical structure γl,k (t) = γl (t − kT ):
Z
c̃l,k = hx ∗ h, γl,k i =
(x ∗ h) (t) γl,k (t) dt.
R
In order to guarantee the stable and effective transmission of the information contained in the sequence
{cl,k } ∈ l2 ({0, . . . , N − 1} × Z), the families {gl (t)}
and {γl (t)} need to satisfy the following conditions:
(i) {gl,k (t)} and {γl,k (t)} are Riesz bases of their
linear span to guarantee stability and continuity
of the synthesis and analysis maps.
A
X
l,k
where
X
2
X
2
2
|cl,k | ≤ cl,k gl,k ≤ B
|cl,k |
l,k
kxk2 =
l,k
Z
R
|x(t)|2 dt.
Note that finite-length, discrete-time signals establish a finite-dimensional Hilbert space. Here,
the notion of a Riesz basis reduces to a set of N
linear independent vectors of dimension N .
(ii) Biorthogonality of the families {gl,k (t)} and
{γl,k (t)}, i.e.,
hgl,k , γl′ ,k′ i = δl,l′ δk,k′ ,
to ensure cl,k = c̃l,k for all l, k, in case of an
ideal channel K = Id.
(iii) The families {gl (t)} and {γl (t)} should be structured to allow for fast synthesis and analysis algorithms.
(iv) Uniform compact support, i.e., supp gl ⊂ [0, T ],
in order to restrict ISI and time delay.
(v) Efficient use of an assigned bandwidth.
(vi) Low ICI/ISI for an ensemble of convolution operators H ⊂ L(L2 (R)), i.e., for all h(t) ∈ H :
Ghl,k,l′ ,k′ := hh ∗ gl,k , γl′ ,k′ i ≈ dl,k δl,l′ δk,k′ .
(2.1)
The most prominent coherent function systems satisfying conditions i) - v) are Weyl–Heisenberg and wavelet
systems which will be described in Section 4 and Section 5. The mentioned fast algorithms are based on the
FFT, and on Mallat’s cascade algorithm, respectively.
For theoretical analysis of the ICI/ISI of MCM pulses,
we consider a set H of absolutely integrable impulse
responses h(t) defined by the following requirements:
(i) The impulse response is supported within a centered interval of length t0 ,i.e.,
supp h ⊆ [− t20 , + t20 ].
Although h(t) does not have finite support in general, we may cut it off at some point and treat the
influence of the remaining part as noise.
(ii) Assume that the first moment of the magnitude–
squared impulse response vanishes (corresponding to properly defined timing recovery):
Z
|h(t)|2 t dt = 0.
R
(iii) The maximum of the magnitude transfer function is normalized (corresponding to perfect automatic gain control)
sup |H(f )| = 1.
f
3. GENERAL RESULTS
To analyze the ICI/ISI of the pulses {gl,k (t)} independently of the analysis pulses {γl,k (t)} we choose orthogonal perturbation as a measure of stability, i.e., we
define
dg,h = Kh g − P<g> (Kh g)L2 ,
where P<g> is the orthogonal projection onto the span
h g,gi
of g(t), given by P<g> (Kh g)(t) = hKhg,gi
g(t) (cf.,
Figure 2).
*
A dg,h −→
p
A
Kh g
A
→
g
Theorem 2 (Lower bound) There exist constants
s ∈ ]0, 1[ and r ∈ [ 12 , 1[ such that for g(t) ∈ L2 (R),
kgkL2 = 1, with supp g ⊆ [α, α + Tg ] for some α ∈ R
and Tg > 0, we have
and
Fig. 2. Orthogonal distortion(=perturbation)
Note that if the family {γl,k (t)} is orthonormal, the orthogonal perturbation of gl,k (t) is related to the ICI/ISI
of the synthesis prototype gl,k (t) via
d2gl,k ,h ≥
X
l′ 6=l,k′ 6=k
|Ghl,k,l′ ,k′ |2 =
X
l′ 6=l,k′ 6=k
|hh∗gl,k , γl′ ,k′ i|2 .
In our treatise, we aim at finding lower and upper bounds
on the orthogonal perturbation dg,h for all h(t) ∈ H.
For simplicity, we define
dg = sup dg,h .
h∈H
Since the convolution Kh g(t) = h∗g(t) corresponds to
multiplication in the Fourier domain, dg can be related
to the frequency localization of g(t), as the following
theorem shows.
Theorem 1 (Upper bound) For g(t) ∈ L2 (R) with
kgkL2 = 1, we have
2
d2g ≤ (πt0 )2 σ|G|
2 ,
Z
b
R
(f −µ)2 |G(f )|2 df
with
µ = µ|G|2 =
Z
b
R
f |G(f )|2 df .
(Note that G(f ) denotes the Fourier transform of g(t)).
On the other hand, one must expect that signals which
are not well localized on the frequency side potentially
undergo a relatively strong orthogonal perturbation.
Clearly, for a given convolution operator there might be
arbitrarily bad localized functions g(t) which are exact
eigenfunctions of this specific operator, so dg,h = 0 for
this particular h(t) — but for practical purposes, we
require a family of basis functions that are stable under
the action of all h(t) ∈ H. Therefore, to be able to
show that certain families are inadequate, we want to
determine a lower bound for dg . Using an uncertainty
principle obtained by Slepian, Pollak, and Landau [11,
12, 13] we obtain the following result.
12
s Tg
for
Tg
t0
≤
1
,
2s
for
Tg
t0
>
1
.
2s
4. SPECIFIC MCM SCHEMES
We now proceed to apply these results to compare structured signal families of wavelet, Wilson, and
Weyl–Heisenberg type with respect to their stability
under convolution operators.
For numerical results based on Theorem 1 and Theorem 2 we assume supp h ⊆ [− t20 , + t20 ] and T = 50 t0 .
.
Furthermore, we will use as realistic choice r = 0.9
.
and s = 1.
As for the number of elements N in the family, N ≥
256 seems realistic; in VDSL applications, N ≈ 2000
is used.
Weyl–Heisenberg or Gabor systems [7] correspond to a
rectangular tiling of the time–frequency plane, the gl (t)
and γl (t) are modulated versions of appropriately chosen prototype function g0 (t) and γ0 (t), i.e.,
gl (t) = g0 (t)e2πi(ρ/T )lt ,
γl (t) = γ0 (t)e2πi(ρ/T )lt .
Existence of Riesz bases requires [10]
ρ ≥ 1.
Thus we have
2
2
where σ|G|
2 is the variance of |G| , i.e.,
2
σ|G|
2 =
4 T
d2g ≥ r2 1 − s g
3
t0
2
1 r t0
d2g ≥
supp gl = supp g0
and
|Gl (f )|2 = |G0 (f − (ρ/T )l|2 .
Since the variance is translation invariant, we have
2
2
σ|G
2 = σ|G |2
0
l|
for all gl (t), so the upper bound from Theorem 1 holds
uniformly in h(t) ∈ H and l = 0 . . . N −1.
Using for g0 (t) a triangle function, a trapezoidal function, or the polynomial t2 (t−a)2 (properly normalized)
yields
.
d2g = 0.0012 .
It is worth emphasizing that the main property ensuring
this uniform upper bound is the fact that within a Weyl–
Heisenberg family, all Gl (f ) share the same frequency
localization.
The real-valued Wilson bases are equivalent to the so–
called OFDM/OQAM systems [2] and can be formu-
lated in terms of sin-components and cos-components:
g0 (t) = g(t) ,
√
gm,1 (t) = g(t) 2 cos(2π 2m
T t),
√
T
gm,2 (t) = g(t− 2 ) 2 cos(2π 2m−1
T t) ,
√
2m−1
gm,3 (t) = g(t) 2 sin(2π T t),
√
gm,4 (t) = g(t− T2 ) 2 sin(2π 2m
T t) ,
where the index set is defined as m ∈ [0, M − 1] and
the corresponding number of pulses per symbol time is
given by N = 4M +1. Similar to WH-Systems, modulation and demodulation can be realized efficiently via
FFT. A recently developed theory allows the design of
pulses g(t) with improved frequency localization [2].
Nevertheless, the following theorem holds.
with respect to a normalized convolution operator reflecting a 2km, 0.4mm PE twisted copper wire cable.
In order to visualize the channel matrices (2.1), we resort the synthesis and analysis families and display a
segment of the biinfinite block Toeplitz matrix.
To compare the families discussed within a setting similar to ADSL, we choose T ≈ 250µs and bases capable
of transmitting about 500 real coefficients (250 imaginary coefficients) utilizing the baseband [-1,1] MHz.
The impulse response of a 2km, 0.4mm PE twisted
copper wire cable sampled at 2 MHz has been calculated according to [6]. The resulting causal impulse
response has been shifted to improve the performance
of all the coherent families discussed here. Additionally, the impulse response has been normalized so that
sup |H(f )| = 1. The resulting function h(t) is shown
in Figure 3.
Theorem 3 In a Wilson basis with at least 200 elements and supp g0 ⊆ [0, T ], there is an element gl with
d2gl ≥
r2
5
y
0.8
.
= 0.16 .
0.6
0.4
The popular dyadic wavelet bases [5, 16] are defined
via
T
(n)
gm
(t) = 2m/2 g0 2m (t−n m ))
0.2
2
where g0 is an appropriate prototype function (referred
to as mother wavelet) and the index set is defined as
follows:
Μs
10
-10
20
30
40
50
-0.2
-0.4
m = 0, 1, . . . , M ,
n = 0, 1, . . . , 2m −1,
(The total number of transmission pulses within symbol period T is accordingly given by N = 2M +1 −1)).
In a dyadic wavelet basis, we encounter the problem
that, since scaling on the time side results in reverse
scaling on the frequency side, the frequency localization gets worse and worse as the indices grow. The
following result gives a quantitative estimate of this effect.
Theorem 4 In a dyadic wavelet family with supp g0 ⊆
[0, KT ] and finest scaling level M ≥ 7 + log2 (K), the
(n)
elements gM (t) on level M satisfy
d2g(n) ≥ 0.81 (1 − 67 · 2−M K) .
M
Fig. 3. Channel impulse response h(t) of a 2km
0, 4mm PE twisted copper wire cable.
The colormap used in the Figures below is shown in 4.
10−6
10−4
10−2
100
Fig. 4. Colormap used in the channel matrix images.
To illustrate the performance of an exemplary Weyl–
Heisenberg system, we choose as prototype functions
1
χ[0,250) (t [µs]).
g0 (t) = γ0 (t) = √
250
When using the orthogonal Daubechies wavelet with
four vanishing moments (db4 in MatLab), we may choose Using T = 250µs, b = 1 MHz and L = 250, we
250
K = 8. If N > 1024, we need M ≥ 11 which yields
obtain the channel matrix displayed in Figure 5. Details
2
d (n) ≥ 0.386; for N > 2048 with M ≥ 12 we obtain
of this matrix are shown in Figure 6.
g11
In the Weyl–Heisenberg example, we shall now employ
d2(n) ≥ 0.598.
g12
a cyclic prefix of 30µs. Hence, the prototype functions
we choose
5. NUMERICAL SIMULATIONS
1
g0 (t) = √
χ[−30,250) (t [µs])
250
To corroborate our theoretical results of Section 3 in a
realistic setup, we shall compute the effective channel
and
matrix of an exemplary wavelet transmission basis and
1
γ0 (t) = √
χ[0,250) (t [µs]).
of an exemplary Weyl–Heisenberg transmission basis
250
50
50
100
100
150
150
200
200
250
50
Fig. 5.
family.
100
150
200
250
Channel matrix of a Weyl–Heisenberg
250
100
150
200
250
Fig. 7.
Channel matrix of a Weyl–Heisenberg
family using a cyclic prefix.
5
5
10
10
15
15
20
20
25
50
25
5
10
15
20
25
5
10
15
20
25
Fig. 6. Details of Figure 5.
Fig. 8. Details of Figure 7.
Using T = 280µs, and L = 250, we obtain the segments of the channel matrix displayed in Figure 7 and
in Figure 8. Note that we do not have exact diagonalization, since the duration of the cyclic prefix is smaller
than the duration of the impulse response h (see Figure 3).
Next, we consider a Wilson basis of dimension 496.
We choose
The prototype function g0 (t) is displayed in Figure 11.
We reorder the orthonormal wavelet family according
to
(n)
g511k+2m −1+n (t) = gk,m (t).
1
g0 (t) = γ0 (t) = √
χ[0,250) (t [µs])
250
with M = 124. The other parameters are chosen as in
the Weyl–Heisenberg example. The channel matrix is
shown in Fig. 9 and Fig. 10.
To obtain an exemplary wavelet system we shall use
the orthogonal Daubechies wavelet with 4 vanishing
moments, i.e., g0 = db4, scaled to support [0, 1791]µs
and normalized in the L2 (R) sense. Furthermore, we
set T = 28 = 256µs and M = 8 and obtain the transmission family
(n)
{gk,m (t)}m=0,1,2,3,....8, n=0,1,....2m −1, k∈Z .
A segment of the resulting effective channel matrix is
displayed in Figure 12. The wavelet basis elements on
the finest scale suffer the strongest orthogonal perturbation, reflecting their poor frequency localization.
Finally we consider a DS spread spectrum system based
on a Walsh-Hadamard code. This modulation scheme
is used in the North American cellular wireless system
[14, p.849]. The transmission signal is given by
XX
x(t) =
ck,l g (l) (t − kT )
k
l
where g (l) (t) is the convolution of the binary-valued
code sequence with index l and some bandlimited pulseshaping signal. We consider this modulation scheme in
a single-user setup, i.e., all of the ck,l contain information bits of the same user. (In the CDMA-type application of DS spread-spectrum l is the user index.) By
y
0.05
50
0.04
100
0.03
150
0.02
0.01
200
Μs
1000
250
2000
3000
4000
5000
6000
7000
-0.01
300
-0.02
350
-0.03
Fig. 11. g0 in the wavelet basis.
400
450
50
100
150
200
250
300
350
400
50
450
Fig. 9. Channel matrix of a OFDM/OQAM (Wilsontype family).
100
150
200
250
5
300
350
10
400
450
15
500
50
100
150
200
250
300
350
400
450
500
Fig. 12. Channel matrix of a wavelet family.
20
25
5
10
15
20
25
5
Fig. 10. Details of Figure 9.
10
the principle of spread-spectrum communication, the
transmission components g (l) (t) deliberately have bad
frequency-localization. Hence one must expect poor
off–diagonal decay. Observe, however, that the robustness advantage of CDMA systems leads to a reduced
variance of the diagonal elements (compared to OFDM
where the diagonal elements correspond essentially to
samples of H(f )).
15
20
25
5
Acknowledgements
Sponsoring by the German Ministry of Education and
Science (BMBF) within the KOMNET program under
grant 01 BP 902 is gratefully acknowledged.
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